1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Prove that the series SUM (-1)^n n/p_n converges where p_n are primes

  1. Mar 5, 2007 #1
    1. The problem statement, all variables and given/known data

    Prove that [tex]\sum(-1)^n\frac{n}{p_n}[/tex] converges, where [tex]p_n[/tex] is the nth prime.

    2. Relevant equations

    The sequence [tex]\frac{n}{p_n}[/tex] is definately not monotone if there exists infinitely many twin primes, since [tex]2n-p_n<0[/tex] for sufficiently large n, so alternating series test is out. Are there any other ways of showing this converges?
     
  2. jcsd
  3. Mar 5, 2007 #2

    StatusX

    User Avatar
    Homework Helper

    Can you use the prime number theorem? This says that:

    [tex] \lim_{n \rightarrow \infty} \frac{p_n}{n \ln n} = 1 [/tex]
     
  4. Mar 10, 2007 #3
    I can't use it for the series. I can only establish that n/p_n -> 0, which is insufficient for the series. I can't even prove that n/p_n is NOT monotone for large n, unless I assume the twin prime conjecture, for example.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Prove that the series SUM (-1)^n n/p_n converges where p_n are primes
  1. Prove n<prime<n! (Replies: 10)

Loading...